Faraz's reciprocal sequence

Consider the sequence {ai}i≥0\{a_i\}_{i\geq0}{ai​}i≥0​ such that a0=7+1a_0=\sqrt{7}+1a0​=7​+1 and an+1=an2−an+1a_{n+1}=a_n^2-a_n+1an+1​=an2​−an​+1. The sum ∑n=0∞1an,\sum_{n=0}^{\infty} \frac{1}{a_n},n=0∑∞​an​1​, can be written as 1b\frac{1}{\sqrt{b}}b​1​. What is the value of bbb?